Contents -- Preface -- Part I -- White Noise Functional Approach to Polymer Entanglements C. C. Bernido and M. V. Carpio-Bernido -- Abstract -- 1 Introduction -- 2 Two Chainlike Macromolecules -- 3 Entanglement with an Intermolecular Potential V (r) -- 4 White Noise Path Integral Approach -- 5 Length-dependent Potentials -- 6 Chirality of Entangled Polymers -- 7 Examples of a Modulating Function -- 7.1 f(s) = k cos (VS) -- 7.2 f(s) = ksp -- 8 Conclusion -- Acknowledgement -- References -- White Noise Analysis, Quantum Field Theory, and Topology A . Hahn -- 1 Introduction -- 2 Chern-Simons models, heuristic path integrals, and topological invariants -- 2.1 Chern-Simons models -- 2.2 Heuristic path integrals and topological invariants -- 2.3 A "Paradox" -- 2.4 The computation of the WLOs: A short overview -- 2.5 Open Questions -- 3 White noise analysis applied to Chern-Simons models on M = R3 in axial gauge -- 3.1 The basic idea -- 3.2 Step 1: Making sense of ...d -- 3.3 Step 2: Making sense of -- 3.4 Step 3: Existence proof and computation of WLO(L -- ) -- 3.5 Comparison with the results obtained by the other approaches -- 4 White noise analysis applied to Chern-Simons models on M = S2 x S1 in torus gauge -- 4.1 Torus Gauge -- 1. Option: -- 2. Option (better): -- 4.2 The Faddeev-Popov-Determinant of Torus gauge fixing -- 4.3 A Formula for Scs(A + Bdt) -- 4.4 Introduction of a scalar product -- 4.5 The decomposition = -- 4.6 A preliminary heuristic formula for WLO(L) -- 4.7 The final heuristic formula for WLO(L) -- 4.8 The Program -- 4.9 Some Details for Step 1 -- 5 Conclusions and Outlook -- References -- A Topic on Noncanonical Representations of Gaussian Processes Y, Hibino -- 1 Introduction -- 2 Noncanonical representations -- Remark -- Acknowledgments. -- References.

Integral Representation of Hilbert-Schmidt Operators on Boson Fock Space U. C. Ji -- 1 Introduction -- 2 Preliminaries -- 2.1 White Noise Triplet -- 2.2 Admissible Triplet -- 3 White Noise Operators -- 3.1 Operator Symbol and Integral Kernel Operators -- 3.2 Chaotic Expansion of Hilbert-Schmidt Operators -- 4 Integral Representation of Hilbert-Schmidt Operators -- 4.1 Quantum Stochastic Integrals -- 4.2 Integral Representation -- 4.3 Integrands in Integral Representation -- References -- The Dawn of White Noise Analysis I. Kubo -- 1. Short history of Brownian motion -- 2. Hida's research before 1975 -- 1. Properties of Paths -- 2. Canonical Representation -- 3. Projective limit of spheres -- 4. Non-linear Analysis -- 5. Infinite Dimensional Rotation Group -- 3. White Noise Analysis -- 4. Comments -- 1. O(c0) -- 2. Sobolev Norms -- 3. Generalized Random Variables -- 4. Renormalization -- 5. Path -- Acknowledgement -- References -- White Noise Stochastic Integration H.-H. Kuo -- 1. Brownian motion and white noise -- 2. White noise theory -- 3. White noise stochastic integrals -- 4. Hitsuda-Skorokhod integrals -- 5. Extensions of Ito's formula -- 6. Stochastic integral equations -- 7. Perspectives of white noise stochastic integration -- (1) Hitsuda-Skorokhod integral -- (2) Solutions of white noise stochastic integral equations -- (3) Extensions of Ito's formula -- (4) Clark-Ocone formula -- (5) Girsanov theorem -- Acknowledgments -- References -- Connes-Hida Calculus and Bismut-Quillen Superconnections R. Leandre and H. Ouerdiane -- I. Introduction -- II. Cyclic homology and white noise analysis -- III. The J.L.O. cocycle as a white noise distribution -- IV. References -- A Quantum Decomposition of Levy Processes Y.-J. Lee and H.-H. Shih -- Abstract -- 1 Introduction -- 2 Levy white noise measure and a representation of Levy processes.

3 Test and generalized Levy white noise functionals -- Segal-Bargmann transform -- Annihilation and creation operators -- 4 Quantum decomposition of Levy processes -- References -- Generalized Entanglement and its Classification T. Matsuoka -- 1. Introduction -- 2. Operational structure of quantum entanglement -- 2.1. Pure state -- 2.2. Classification of states via entangling operator -- 3. Degree of entanglement via quantum mutual entropy -- 3.1. Characterization of a pure state by degree of entanglement -- 3.2. Degree of entanglement for entangled PPT states -- References -- A White Noise Approach to Fractional Brownian Motion D. Nualart -- 1 Introduction -- 2 White noise analysis and Hida distributions -- 3 Derivative of a Volterra process -- 4 Fractional Brownian motion -- 5 Stochastic calculus with respect to the fBm -- 5.1 Stochastic calculus of variations with respect to fBm -- 5.2 Divergence and symmetric integrals for H > -- 5.3 Ito's formula for the divergence integral for H > -- 5.4 Stochastic integration with respect to fBm for H < -- 6 White noise analysis and divergence integrals -- References -- Adaptive Dynamics in Quantum Information and Chaos M. Ohya -- 1 Introduction -- 2 Adaptive Dynamics -- 3 Entropic Chaos Degree -- 4 Algorithm Computing Entropic Chaos Degree -- 4.1 Logistic Map -- Tinkerbell map -- 4.2 ECD with memory -- 5 Description of Chaos by Adaptive Dynamics -- 5.1 Chaos degree with adaptivity -- References -- Micro-Macro Duality in Quantum Physics I. Ojima -- Abstract -- 1 Why & what is Micro-Macro Duality? -- 2 Basic scheme for Micro/Macro correspondence -- 2.1 Definition of sectors and order parameters -- 2.2 Selection criteria to choose an appropriate family of sectors -- 3 Sectors and symmetry: Galois-Fourier duality -- 3.1 Hierarchy of symmetry breaking patterns and augmented algebras.

4 From [thermality geometry] towards [history of Nature] -- References -- White Noise Measures Associated to the Solutions of Stochastic Differential Equations H. Ouerdiane -- Mathematics Subject CIassifications (2000): 60H40 -- 1 Introduction -- 2 Preliminaries -- 3 Application to white noise analysis -- 3.1 S-Transform -- 3.2 Relation of this theorem with previous results -- 4 Convolution calculus -- 4.1 Convolution operators -- 4.2 Symbols of operators -- 4.3 Convolution product of operators -- 5 Applications to stochastic differential equations -- 5.1 Asymptotic estimates for white noise measures -- 5.2 Tail estimates for solutions of stochastic differential equations -- References -- A Remark on Sets in Infinite Dimensional Spaces with Full or Zero Capacity J. Ren and M. Rockner -- Abstract -- 1. INTRODUCTION, FRAMEWORK AND A RESULT ON SETS WITH FULL CAPACITY -- 2. PROOF OF THEOREM 1.4 -- 3. A RESULT ON SETS WITH ZERO CAPACITY -- ACKNOWLEDGEMENT -- REFERENCES -- An Infinite Dimensional Laplacian in White Noise Theory K. Saitd -- Mathematics Subject CIassifications (2000): 60H40 -- Introduction -- 1 White noise functionals of Gaussian noise and Poisson noise -- 2 The Levy Laplacian acting on white noise functionals -- 3 The Levy Laplacian acting on WNF-valued functions -- 4 An infinite dimensional stochastic process associated with the Levy Laplacian -- Acknowledgments -- References -- Invariance of Poisson Noise Si Si, A. Tsoi and Win Win Htay -- 1. Introduction -- 2. Preliminaries -- 3. Invariance of p under transformations on parameter space -- 3.1. Transformations in the Rd-parameter case -- 3.2. Td-parameter case -- 4. Transformations that preserve conditional Poisson measures -- 5. A characterization of Poisson noise -- References -- Nonequilibrium Steady States with Bose-Einstein Condensates S. Tasaki and T. Matsui -- 1 Introduction.

2 Bosonic Junction System -- 3 Nonequilibrium Steady States -- 4 Boson on Lattice -- 5 NESS -- (a) Free Boson. -- (b) Impurity Scattering -- (c) Two-Dimensional Layer -- Acknowledgments -- Appendix -- References -- Multidimensional Skew Reflected Diffusions G. Trutnau -- Summary. -- 1 The generalized Dirichlet form and its capacity -- 2 Construction of an associated diffusion -- 3 The Revuz correspondence -- 4 Semimartingale characterization -- 5 Identification of the process -- 6 Recurrence -- References -- On Quantum Mutual Type Entropies and Quantum Capacity N . Watanabe -- 1. Quantum channels -- 1.1. Noisy optical channel -- 2. Ohya S-mixing entropy and Ohya mutual entropy -- 3. Comparison of various quantum mutual type entropies -- 4. Quantum capacity -- References -- Part II -- White Noise Calculus and Stochastic Calculus L. Accardi and A. Boukas -- 1. Introduction -- 2. Elemental processes -- 3. White noise approach to classical and quantum stochastic calculus -- 4. Nonlinear powers of white noise -- 5. Emergence of the square of white noise in different contexts -- 6. Higher powers of white noise -- 7. No go theorems -- Appendix: Meixner's classification theorem -- 8. Orthogonal generating functions -- 9. The equations for f and for 21 = 21-l -- 10. Meixner's parametrization -- 11. Solutions of the equation for v -- 12. Solutions of the equation for f -- 13. The equations for u -- 14. Moments of the Meixner measures -- 15. Bibliography.